The Irish alphabet doesn't actually contain the letter H, even though it appears constantly in modern Irish spelling! The H is used to denote a special effect called lenition -- which is really a fancy way of talking about aspiration of consonants.

The H is used in modern Irish spelling -- in older Irish (called Ogham script), the same effect was noted by putting a dot over the letter. As you can imagine, writing with all sorts of lines and dots and acutes sporting about with the letters got pretty confusing to read. Hence, they introduced the convention of using H to show aspiration (lenition) in written Irish.

The alphabet also does not contain a few other letters (although they'll often show up in loan words adopted into Irish.)

a b c d e f g i l m n o p r s t u

Lenition changes the way a letter is pronounced, and is shown by adding -h- after the initial letter of the word. In Irish, this occurs when the word is used in a specific way, or follows certain words. For example, the general article 'the'', as in 'the carr' or 'the woman', causes lenition in most words. Lenition is called séimhiú in Irish. For example:

bean (woman) — an bhean (the woman)

Normally, words beginning with the letters l, n , r are not changed. The following letters can be affected:

b c d f g m p s t — become— bh ch dh fh gh mh ph sh th

However, words beginning with sc-, sf-, sm-, sp-, and st- do not change.

Another change that can occur to words in Irish is eclipsis. Like lenition, it is caused by the preceding word and it can effect both consonants and vowels (unlike lenition, which affects only consonants). Eclipsis is called urú in Irish. Eclipsis adds another letter that replaces the sound of the original letter. When you see the following combinations, pronounce the first letter and ignore the second -- it's left in place simply to show you what the word looked like before. Note that bh is considered a single letter!

b c d f g p t — become— mb gc md bhf ng bp dt
a e i o u — become— n-a, n-e, n-i, n-o, n-u

Unlike English, Irish does have gender, and each noun is either masculine or feminine. Whether the word is masculine or feminine determines how the word is changed. All dictionaries note whether a noun is masculine or feminine (and you can often guess, for example, fear (man) is masculine, while bean (woman) is feminine, as expected.)